Hausdorff Gap

In mathematics, a Hausdorff gap consists roughly of two collections of sequences of integers, such that there is no sequence lying between the two collections. The first example was found by . The existence of Hausdorff gaps shows that the partially ordered set of possible growth rates of sequences is not complete.

Definition

Let $\omega^\omega$ be the set of all sequences of non-negative integers, and define f to mean $\lim \left (g(n)-f(n)\right )=+\infty$.

If $X$ is a poset and $\kappa$ and $\lambda$ are cardinals, then a $(\kappa,\lambda)$-''pregap'' in $X$ is a set of elements $f_\alpha$ for $\alpha\in\kappa$ and a set of elements $g_\beta$ for $\beta\in\lambda$ such that:
*The transfinite sequence $f$ is strictly increasing;
*The transfinite sequence $g$ is strictly decreasing;
*Every element of the sequence $f$ is less than every element of the sequence $g$.

A pregap is called a ''gap'' if it satisfies the additional condition:

*There is no element $h$ greater than all elements of $f$ and less than all elements of $g$.

A Hausdorff gap is a $(\omega_1,\omega_1)$-gap in $\omega^\omega$ such that for every countable ordinal $\alpha$ and every natural number $n$ there are only a finite number of $\beta$ less than $\alpha$ such that for all $k>n$ we have f_\alpha(k).

There are some variations of these definitions, with the ordered set $\omega^\omega$ replaced by a similar set. For example, one can redefine f to mean f(n) for all but finitely many $n$. Another variation introduced by is to replace $\omega^\omega$ by the set of all subsets of $\omega$, with the order given by A if $A$ has only finitely many elements not in $B$ but $B$ has infinitely many elements not in $A$.

Existence

It is possible to prove in ZFC that there exist Hausdorff gaps and $(b,\omega)$-gaps where $b$ is the cardinality of the smallest unbounded set in $\omega^\omega$, and that there are no $(\omega,\omega)$-gaps. The stronger open coloring axiom can rule out all types of gaps except Hausdorff gaps and those of type $(\kappa,\omega)$ with $\kappa \geq \omega_2$.

References

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External links

*{{eom, title=Hausdorff gap

Descriptive set theory
Order theory
Integer sequences
General topology